PX157 - E4c - LRC circuits 3

step 1:

{\tilde{Z}}_{e q} = R + j ω L + \frac{1}{j ω C}

step 2:

\tilde{I} = \frac{{\tilde{V}}_{i n}}{{\tilde{Z}}_{e q}} = \frac{{\tilde{V}}_{i n}}{R + j (ω L - \frac{1}{ω C})}

step 3:

\begin{aligned} {\tilde{V}}_{o u t} & = ({\tilde{Z}}_{L} + {\tilde{Z}}_{C}) \tilde{I} \\ = (j ω L + \frac{1}{j ω C}) \tilde{I} \\ = \frac{j (ω L + \frac{1}{j ω C})}{R + j (ω L - \frac{1}{ω C})} {\tilde{V}}_{i n} \end{aligned}

step 4:

\frac{| {\tilde{V}}_{o u t} |}{| {\tilde{V}}_{i n} |} = \frac{| ω^{2} L C - 1 |}{\sqrt{ω^{2} R^{2} C^{2} + (ω^{2} L C - 1)^{2}}}

step 5:

ϕ = \arctan (\frac{ω L - \frac{1}{ω C}}{R})

${\tilde{V}}_{i n}$ leads $\tilde{I}$ by a phase, $ϕ$
this circuit acts as a notch filter
since ${\tilde{V}}_{i n} = {\tilde{V}}_{o u t} + {\tilde{V}}_{R}$ , the voltage drop over the resistor makes a bandpass filter